where \((\Gamma /\sigma )^{\operatorname{st}}=(\Gamma ^{\operatorname{st}}/\sigma )\)  is the stabilization of \(\Gamma /\sigma \) , see Definition REF .
In Theorem REF , we show that \(\mathcal {J}_{g,n}^{trop}\)  is a cone stack and that \(\Phi ^{trop}\)  is a morphism of cone stacks, in the sense of [1]}.
Moreover, \(\mathcal {J}_{g,n}^{trop}\)  admits the cone substacks \(\mathcal {J}_{g,n, d}^{trop}\)  (resp. \(\mathcal {J}_{g,n}^{trop, \operatorname{spl}}\) , resp. \(\mathcal {J}_{g,n}^{trop}(\phi )\)  for any \(\phi \in V_{g,n}\) ) parametrizing pairs
\((\Gamma /\sigma ,D)\)  such that \(\deg D=d\)  (resp. \((\Gamma )\)  is simple in the sense of Definition REF , resp. \(D\)  is \(\phi \) -semistable in the sense of Definition REF (REF )).
